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THE REV. H. MOSELEY ON THE GEOMETRICAL FORMS 



It was made from the dark line which shows, on the section of the internal whorls 

 of the shell, the line of that pearly surface which the animal deposits as a covering to 

 its completed portion, as it advances in the construction of it. It is important to 

 make this observation, because as it extends one whorl of its shell over another, the 

 animal deposits continually upon the pearly surface of this last a new coating of 

 shell, and thickens it ; and it is in the centre of this thickened section that is to be 

 found that section of the pearly surface, of which the edge of the external whorl is 

 a continuation, and from which this tracing was taken. 



It will be found that the distance of any two of its whorls measured upon a radius 



vector is one-third that of the two next whorls measured upon the same radius vector. 



Thus 



ah is one-third of h c, 



«? e is one-third of ef, 



. g h\^ one-third of h i, 



Ik is one-third of Im. 



The curve is therefore a logarithmic spiral. 



Turbo duplicatus. 



From the apex of a large specimen of the Turbo duplicatus a line was drawn across 

 its whorls, and their widths were measured upon it in succession, beginning from the 

 last but one. The measurements were, as before, made with a fine pair of compasses 

 and a diagonal scale. The sight was assisted by a magnifying glass. 



In a parallel column to the admeasurements are the terms of a geometric progres- 

 sion, whose first term is the width of the widest whorl measured, and whose common 

 ratio is 1-1804. 



Yet further to verify this remarkable coincidence of the widths of successive whorls 

 with the mathematical law of a geometric progression, the following property of such 

 a progression was determined: " that fju representing the ratio of the sum of every even 

 number (m) of its terms to the sum of half that number of terms, the common ratio 

 (r) of the series is represented by the formula 



r = (i.-ir: 



